# Thread: Matrix Initial Value Problem

1. ## Matrix Initial Value Problem

Hi, I'm stuck on solving this problem...
Consider the initial value problem



Determine the solution as a function of .

So this is what I did so far
Sorry I dont know how to type matrix on this....

| 3.5 -0.75| * |x1|
|3 -1.5| |x2|

|3.5-
λ -0.75|
|3 -1.5-λ|

λ2 - 2λ - 3 = 0
λ = -1
λ = 3

λ=1
|3.5-1 -0.75|
|3 -1.5-1 |

|2.5 -0.75|
|3 0.5|

This is where I'm stuck...I dont know what to do after

same for
λ=3
|0.5 -0.75|
|3 -4.5|

Thanks!

2. ## Re: Matrix Initial Value Problem

Originally Posted by JC05
Hi, I'm stuck on solving this problem...
Consider the initial value problem



Determine the solution as a function of .

So this is what I did so far
Sorry I dont know how to type matrix on this....

| 3.5 -0.75| * |x1|
|3 -1.5| |x2|

|3.5-
λ -0.75|
|3 -1.5-λ|

λ2 - 2λ - 3 = 0
λ = -1
λ = 3

λ=1
|3.5-1 -0.75|
|3 -1.5-1 |

|2.5 -0.75|
|3 0.5|

This is where I'm stuck...I dont know what to do after

same for
λ=3
|0.5 -0.75|
|3 -4.5|

If your eigensystem is $\large \{ (\lambda_1, \vec{v_1}), (\lambda_2, \vec{v_2}) \}$ then the solution to the system of diff eqs is given by
$\large \begin{pmatrix}x_1(t) \\ x_2(t)\end{pmatrix} = c_1 \vec{v_1} e^{\lambda_1 t} + c_2 \vec{v_2} e^{\lambda_2 t}$
and you use the initial conditionals to solve for $c_1$ and $c_2$.